Abstract
The rowmotion action on order ideals or on antichains of a finite partially ordered set has been studied (under a variety of names) by many authors. Depending on the poset, one finds unexpectedly interesting orbit structures, instances of (small order) periodicity, cyclic sieving, and homomesy. Many of these nice features still hold when the action is extended to [0,1]-labelings of the poset or (via detropicalization) to labelings by rational functions (the birational setting).
Highlights
Combinatorial rowmotion is a well-studied action on the set of order ideals J (P ) or on the set of antichains A(P ) of a finite poset P
Rowmotion has proven to be of great interest in dynamical algebraic combinatorics
We show that NOR-motion (Noncommutative Order Rowmotion) and NAR-motion (Noncommutative Antichain Rowmotion) always exhibit the same order on any given finite poset
Summary
Combinatorial rowmotion is a well-studied action on the set of order ideals J (P ) or on the set of antichains A(P ) of a finite poset P. The first author gave an explicit isomorphism between these two different toggle groups (on J (P ) and on A(P )) for the same poset P , and extended these results to the piecewise-linear level [12], where A(P ) extends to Stanley’s chain polytope C(P ) [23] These toggles can be used to define the antichain rowmotion of [2] and its extension to all of C(P ). We have four realms (combinatorial, piecewise-linear, birational, and noncommutative) and two rowmotion maps (order-ideal and antichain). We discuss the toggle group of a poset P , rowmotion on order ideals and on antichains, and define their generalizations to the piecewise-linear realm. The toggling perspective allows us to extend these maps from the combinatorial realm (on finite sets) to the piecewise-linear realm (polytopes whose vertices correspond to these sets), and lift to the birational realm by detropicalizing the operations [6]. An example of each of the birational maps ∇, ∇−1, ∆, and ∆−1 on the positive root poset Φ+(A3)
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